In the literature of run length limited coding, the Repeated Maximum Transition Run, i.e. RMTR, constraint is often referred to as the MTR constraint. Originally, the maximum transition-run (MTR) constraint as introduced by J. Moon and B. Brickner, in “Maximum transition run codes for data storage systems”, IEEE Transactions on Magnetics, Vol. 32, No. 5, pp. 3992-3994, 1996, (for a d=0 case) specifies the maximum number of consecutive “1”-bits in the NRZ bitstream (where a “1” indicates a transition in the related bi-polar channel bitstream). Equivalently, in the (bi-polar) NRZI bitstream, the MTR constraint limits the number of successive 1T runs. As argued above, the MTR constraint can also be combined with a d-constraint, in which case the MTR constraint limits the number of consecutive minimum runlengths, as is the case for the 17PP code which is used in the Blu-Ray disc (BD) format. The basic idea behind the use of MTR codes is to eliminate the so-called dominant error patterns, that is, those patterns that would cause most of the errors in the partial response maximum likelihood (PRML) sequence detectors used for high density recording. A highly efficient rate 16→17 MTR code limiting the number of consecutive transitions to at most two for d=0 has been described by T. Nishiya, K. Tsukano, T. Hirai, T. Nara, S. Mita, in “Turbo-EEPRML: An EEPRML channel with an error correcting post-processor designed for 16/17 rate quasi MTR code”, Proceedings Globecom '98, Sydney, pp. 2706-2711, 1998. Another argument in favor of the RMTR constraint is to limit the back-tracking depth (or trace-back depth) of the Viterbi (PRML) bit-detector. The disclosure of U.S. Pat. No. 5,943,368 aims to encode data into a channel bitstream that prohibits the generation of single-frequency components (which can be a long repetition of (minimum) runlengths).
The RMTR constraint has recently regained some interest in the optical recording community. The ETM-code disclosed in K. Kayanuma, C. Noda and T. Iwanaga, “Eight to Twelve Modulation Code for High Density Optical Disk”, Technical Digest ISOM-2003, Nov. 3-7, 2003, Nara, Japan, paper We-F-45, pp. 160-161 has d=1, k=10 and r=5 constraints, this r constraint being just one lower than the RMTR of 17PP. For d=1 and RMTR r=2, the theoretical Shannon capacity amounts to:C(d=1,k=∞,r=2)=0.679286.  (1)
So, a code with rate better than ⅔ is still feasible. For an even more aggressive RMTR constraint r=1, the theoretical Shannon capacity amounts to:C(d=1,k=∞,r=1)=0.650900.  (2)This shows that r=2 is the lowest RMTR constraint that is possible for a code rate not lower than that of the 17PP code.
Recently, in K. A. S. Schouhamer Immink, “Method and Apparatus for Coding Information, Method and Apparatus for Decoding Coded Information, Method of Fabricating a Recording Medium, the Recording Medium and Modulated Signal”, PCT Patent WO 02/41500 A1, International Filing Date 11 Nov. 2000, and in K. A. S. Immink, J.-Y. Kim, S.-W. Suh, S. K. Ahn, “Efficient dc-Free RLL Codes for Optical Recording”, IEEE Transactions on Communications, Vol. 51, No. 3, pp. 326-331, March 2003, some very efficient d=1 codes were disclosed with a code-rate that is very close to the Shannon capacity for d=1, given by C(d=1, k=∞, r=∞)=0.6942. As an example, a code with a rate of R= 9/13 has been realized, which has a code efficiency
  η  =      R    C  such that 1−η=0.28%. However, these very efficient RLL codes suffer from the absence of an RMTR constraint (r=∞); therefore, the latter 9-to-13 d=1 code cannot yield the practical capacity benefit of 5% (through adapted PRML sequence detection) that is offered by d=1 codes with r=2.
The performance gain due to the RMTR constraint has been studied experimentally for high-density optical recording channels derived from the Blu-ray Disc (BD) system. Experiments have been performed using the increased-density BD rewritable system with the disc capacity increased from the standard 23.3-25-27 GB to 35 GB. PRML (Viterbi) bit detection has been employed.
Performance of the Viterbi bit detector has been measured based on the sequenced amplitude margin (SAM) analysis. In the relevant range of capacities around 35 GB, 1 dB gain in SAMSNR means almost 6% disc capacity increase.
Channel codes with different RMTR constraints have been compared to each other. In order to separate read-channel performance gain due to the imposed RMTR constraint from the corresponding write-channel gain, two different Viterbi bit detectors have been used: one which is aware of the RMTR constraint, and the other which is not. In the second case the performance gain can be attributed solely to the improved spectral content of the data written on the disc (such that it is better matched to the characteristics of the write channel used).
When the 17PP channel code with the RMTR constraint r=6 (as used in the BD system) is employed, SAMSNR of 11.66 dB is achieved for both RMTR-aware and RMTR-unaware bit detectors, i.e. no RMTR-related performance gain is observed in the read channel. When the channel code with r=2 is used, SAMSNR of 12.55 dB and 12.07 dB are achieved for the RMTR-aware and RMTR-unaware bit detectors correspondingly. As one can see, a total RMTR-related SAMSNR increase of about 0.9 dB is gained with respect to the case of r=6, which corresponds with about 5% disc capacity increase.